UNIVERSITY OF WAIKATO

Department of Mathematics

MATH509-16A Number Theory 2016

Paper Outline


A two hour bookwork test will be on Monday 20th June at 2pm in G3.20 (off G3.19). Assignments to be marked should be handed in by Friday 10th June 5pm.

web site: math.waikato.ac.nz/~kab/math509-16A.html

Lecture Notes:

  • Set 1: Notation, Euler summation, Stirling's formula, steepest descent, Abel summation, Chinese remainder theorem, density of sets of integers.
  • Set 2: Arithmetic-geometric mean, Euler summation, Mersenne primes, Fermat primes, lower bounds for pi(x), conjectures for primes.
  • Set 3: The Riemann zeta function, infinite products being non-zero, zeta non-zero for Re s>1.
  • Set 4: Chebychev's approximate prime number theorem, Bertrand's hypothesis.
  • Set 5: The arithmetic functions psi(x), theta(x), mu(n), and Lambda(n).
  • Set 6: General Mobius inversion, the series mu(n)/n converges to zero, start of the PNT.
  • Set 7: Lecture notes for a first course in number theory (Do Not Print all this - its 11Mbytes!!!!!! we will cover only some parts.)
  • Set 8: Global behaviour of arithmetic functions, sum of divisors and Euler phi functions.

    Rationale:

    For some time now there has been developing within and outside of mathematics a renewed energy and interest in matters relating to number theory. This has come from not only the recent solution to some unsolved long outstanding problems using modern methods, like Wilesí proof of Fermats Last Theorem, but also because of the pressing need for effective encryption in commercial, strategic and personal computer based communications. In addition, the use of the computer has made it possible to explore a much wider domain of number based phenomena than before, leading to new ideas. However, quite a few classical problems relating to the fundamental structure of natural numbers, are still unsolved. Since these numbers underpin all of mathematics and its applications, there is a lot of challenge and life in this subject.
    During even the past three years there have been remarkable advances in prime number theory. The first was the existence of arithmetic progressions consisting only of primes of any given prescribed finite length - proved by Terry Tao and Ben Green. Then the existence of an infinite set of pairs of successive primes at distance less than or equal to 246, by Zhang and a host of others including Maynard. Finally by Harald Helfgott, the final solution to Goldbach's weak problem - every odd integer can be expressed as the sum of three primes.
    Natural numbers exhibit a high degree of randomness when it comes to their prime factors. For example successive natural numbers have no primes in common! It is the method of analytic number theory to discover properties of infinite subsets of these numbers which have properties we can easily understand and compare. For example that the number of primes is infinite or that the number of primes up to x is like x/log x. Another is that there is always a prime between n and 2n.

    Details of the paper content:

    The following is a list of the type of topics which might be included, but it is not exhaustive and all topics listed would not necessarily be covered:
    1. Summation flormulae, Euler and Abel summation and inequalities, Stirling's formula
    2. Bounds for the number of primes up to x.
    3. The Riemann zeta function, zeta zeros, the Riemann hypothesis
    4. The proof of the prime number theorem
    5. Dirichlet characters
    6. The proof of Dirichlet's theorem on primes in arithmetic progressions
    7. The global and local behaviour of arithmetic functions, average order and normal order
    8. If time permits, smooth numbers and their applications
    9. A summary of recent results on primes in arithmetic progressions, gaps between primes, and Goldbach's conjecture.

    Bibliography:

    There are a number of texts, most of which should be in the library which should be suitable for additional reading. There are quite a few on-line sources. Wikipedia is good for checking definition and some examples.
    1. Terry Tao's analytic number theory 254A announcement and his Notes 1: Elementary Multiplicative Number Theory (Google them)
    2. An introduction to the Theory of Numbers G H Hardy and E M Wright (Oxford 1938-1999)
    3. Introduction to Analytic Number Theory, T M Apostol (Springer Verlag 1976)
    4. Not always burried deep, P Pollack (AMS)
    5. Elementary methods in number theory, M Nathanson (Springer)

    Web Sources:

  • Number Theory Web,
  • Pages on primes,
  • History of Mathematics,
  • Cryptography,
  • Fermats Last Theorem.

    Lecturer:

    Lecture Timetable:

    1. Monday - 4.10-5pm in G3.33 (tutorial/workshop commencing 7th March)
    2. Wednesday noon-1.50pm in G3.33,
    3. Thursday - 1.10pm-2pm in G.3.33

    Prescribed Text:

  • Analytic Number Theory: exploring the anatomy of integers, by Jean-Marie De Koninck and Florian Luca, AMS 2012.

    Kevin Broughan, 3rd June 2016.